As a last check, I'd try using the Finder to burn a DVD. This Apple document explains how to do it. If it fails, I believe that the drive is likely in need of replacement, with a small chance that the connectors just need to be reseated tightly.

I've replaced DVD drives twice before on tower machines. That was relatively easy. A laptop or iMac is more difficult so it might be easier to have a shop do it for you.

I don't have a drive recommendation. I used Pioneer because at the time, that is what Apple was using. They have since mixed up what drives they use.

If my laptop ever needs a replacement, I'll order online since I have the confidence to do the work my self. Good sites like Macsales.com, in the U.S., allow you to choose your computer model and they display only the drives they sell that will work with your model. If you order from such a place, what out for notices like this one; 'For pre-"Unibody" models only!'. You really want to be sure you are buying a drive that fits properly.

The Apple store will use what ever brand they use. If your system is still under warranty, then it will be free.

The difficulty of swapping the drive is dependent on the model. You can probably find guides online that will help you out.

You sound like your are inexperienced with working inside of a computer, so your best option is probably to have the Apple store (or third party Apple representative) take a look at your system and fix it. They can be more thorough about the testing. For instance, the drive is connected to the motherboard, which may be where the problem is.